Abelian fibrations and SYZ mirror conjecture

Cristina Martínez

doi:10.1016/j.crma.2012.07.011

Geometry

Abelian fibrations and SYZ mirror conjecture

Presented by: Mikhaël Gromov

Cristina Martínez ^1,²

¹ Universitat, Departament de Matematiques, Edifici C, Facultad de Ciencies, 08193 Bellaterra, Barcelona, Spain
² Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università di Roma, P.le Aldo Moro, 5, 00185 Roma, Italy

Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 689-692.

Abstract
Résumé

SYZ mirror conjecture predicts that a Calabi–Yau manifold X consists of a family of tori which are dual to a family of special Lagrangian tori on the mirror dual manifold $\hat{X}$ . Here we consider a fibration of polarized abelian varieties and we construct a dual one. Moreover we prove that they are equivalent at the level of derived categories.

La conjecture de « symétrie miroir SYZ » prédit quʼune variété de Calabi–Yau X consiste en une famille de tores qui sont duaux dʼune famille de tores lagrangiennes spéciaux dans la variété miroir duale $\hat{X}$ . Nous considérons ici une fibration de variétés abéliennes polarisées et nous en construisons la duale. De plus, nous montrons quʼelles sont équivalentes au niveau des catégories dérivées.

Received: 2010-04-09
Accepted: 2012-07-25
Published online: 2012-07-01

DOI: 10.1016/j.crma.2012.07.011

Author's affiliations:

Cristina Martínez ^{1, 2}

¹ Universitat, Departament de Matematiques, Edifici C, Facultad de Ciencies, 08193 Bellaterra, Barcelona, Spain
² Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università di Roma, P.le Aldo Moro, 5, 00185 Roma, Italy

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     author = {Cristina Mart{\'\i}nez},
     title = {Abelian fibrations and {SYZ} mirror conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {689--692},
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     volume = {350},
     number = {13-14},
     year = {2012},
     doi = {10.1016/j.crma.2012.07.011},
     language = {en},
}

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Cristina Martínez. Abelian fibrations and SYZ mirror conjecture. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 689-692. doi : 10.1016/j.crma.2012.07.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.07.011/

References
Cited by

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[2] R. Dijkgraaf Mirror symmetry and elliptic curves, Texel Island, 1994 (Progr. Math.), Volume vol. 129, Birkhäuser, Boston, MA (1995), pp. 149-163

[3] R. Donagi; T. Pantev Torus fibrations, gerbes, and duality (in Memoirs of the AMS) | arXiv

[4] D. Henández Ruipérez; A.C. López Martín; F. Sancho de Salas Relative integral functors for singular fibrations and singular partners, J. Eur. Math. Soc. (JEMS), Volume 11 (2009), pp. 597-625

[5] C. Martínez; A.N. Todorov Derived equivalences of Calabi–Yau fibrations | arXiv

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